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Suppose $K$ is a number field, $r=r_1+r_2-1$ (the rank of the unit group) and $u_1,\dots, u_r$ are a basis of fundamental units. Suppose this basis minimizes the max over $i=1,\dots, r$ of the largest archimedean absolute value of $u_i$, and this value is $M>0$. What is a good upper bound on $M$ in terms of standard invariants of the field $K$? (In fact, I know one exists which for fixed degree is exponential in the regulator.)

More precisely, I'd like a bound on this minimax (min over choices of basis of max over basis elements) of the product of all archimedean absolute values which are greater than $1$ (a.k.a. the "entropy") of the $u_i$. (This can of course be bounded by an $r-1$st power of the previous bound.)

Is there a better bound/standard reference for bounding these quantities?

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  • $\begingroup$ Clarify your "more precise" request: all arch. abs. values of all units in a basis or just one unit in a basis which has the largest product or something else? (Side comment: maybe the Mahler measure would be something to look at.) $\endgroup$
    – KConrad
    Commented Aug 22, 2010 at 0:26
  • $\begingroup$ I think that's what he means: $\min_{U} \max_{u_i \in U} {M(u_i)}$, where $U$ ranges over fundamental bases and $M(a)$ is the Mahler measure of $a$. In the real quadratic case, this is just $e^R$, and I guess this bound might be correct for all number fields. $\endgroup$
    – Dror Speiser
    Commented Aug 22, 2010 at 6:13
  • $\begingroup$ Yes, I mean basically what Dror said, namely $min_U max_{u_i\in U}H(u_i)$ (or exp of that), where $U$ ranges over fundamental bases and $H=M^{1/n}$ is the Weil height (with $n=deg(K)$). A bound close to $e^R$ would be nice -- the bound I know is on the order of $e^{n^n R}$. $\endgroup$
    – Dmitry Vaintrob
    Commented Aug 22, 2010 at 22:37
  • $\begingroup$ Correction to the previous comment: yes, I mean precisely the Mahler measure M (not the Weil height H and not its exp - the paper I looked up had renormalized valuations.) $\endgroup$
    – Dmitry Vaintrob
    Commented Aug 22, 2010 at 22:44

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This is a comment, but I can't post any as such...

In case $r = 1$, there is only one fundamental unit, whence your minimax is exponential in the regulator. So something better is only possible in case $r > 1$.

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  • $\begingroup$ You have enough points to make comments. Does your browser run javascript? $\endgroup$
    – S. Carnahan ♦
    Commented Jan 9, 2011 at 12:13
  • $\begingroup$ I didn't had enough points back then :) $\endgroup$
    – felix
    Commented Jan 14, 2011 at 8:46

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